Intro to Analysis in-class_Part_8

# Intro to Analysis in-class_Part_8 - 1.2 Inﬁnite Sets 21...

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Unformatted text preview: 1.2 Inﬁnite Sets 21 Example 1.2.2. Suppose A = {1, 2, 3, 4, 5}. Then the subset {2, 3, 5} corresponds to (0, 1, 1, 0, 1) ∈ 2A . This is the function 1 → 0, 1.2.3 Exercises: #1,3 2 → 1, 3 → 1, Recommended: #2,4 4 → 0, 5→1 Due: Jan. 29 1. Every subset of N is either ﬁnite or countable. 2. If A1 , A2 , A3 , . . . An are countable then n k=1 Ak is countable. 3. Show that the set of algebraic numbers is countable. A number x is algebraic iﬀ a0 + a1 x + a2 x2 + · · · + an xn = 0, for some integers ai . Hint: for N ∈ N, there are only ﬁnitely many equations with n + |a0 | + · · · + |an | = N. 4. Is the set of all irrational real numbers countable? 22 Math 413 1.3 Preliminaries The rational numbers The evolution of numbers ... N = {1, 2, 3, 4 . . . }. To solve equations like x + 5 = 2, need to add negatives: Z = {. . . , −2, −1, 0, 1, 2, . . . }. To solve equations like x × 5 = 2, need to add rationals: Q={ m. . . m, n ∈ Z, n = 0}. n To solve equations like xn = 2, need to add roots like √ n 2. What if we add solutions to n all polynomials an x + . . . a1 x + a0 = 0? We get A with π ∈ A, / but √ −1 ∈ A . So what is R anyway? Roughly: . . R ≈ {lim xn . {xn } ⊆ Q, {xn } converges}. Problem: how to deﬁne lim xn only in terms of Q? The usual deﬁnition of limit says that lim xn = L iﬀ ∀ε > 0, ∃N, n ≥ N =⇒ |xn − L| < ε. This is circular: cannot use a real number L to deﬁne itself. Cauchy sequences will overcome this. 1.3.1 The abstract structure of Q. Deﬁnition 1.3.1. A group is a set with an associative binary operation, an identity, and inverses. Written additively, (G, +, 0) must satisfy 1. x, y ∈ G =⇒ x + y is a well-deﬁned element of G. 2. ∃!0 such that x + 0 = 0 + x = x, ∀x ∈ G. ...
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Intro to Analysis in-class_Part_8 - 1.2 Inﬁnite Sets 21...

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